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Chromatic Graph Theory
DISCRETE MATHEMATICS ITS APPLICATIONS Series Editor Kenneth H. Rosen, Ph.D. Juergen Bierbrauer, Introduction to Coding Theory Francine Blanchet-Sadri, Algorithmic Combinatorics on Partial Words Richard A. Brualdi and Drago˘s Cvetkovi´c, A Combinatorial Approach to Matrix Theory and Its Applications Kun-Mao Chao and Bang Ye Wu, Spanning Trees and Optimization Problems Charalambos A. Charalambides, Enumerative Combinatorics Gary Chartrand and Ping Zhang, Chromatic Graph Theory Henri Cohen, Gerhard Frey, et al., Handbook of Elliptic and Hyperelliptic Curve Cryptography Charles J. Colbourn and Jeffrey H. Dinitz, Handbook of Combinatorial Designs, Second Edition Martin Erickson and Anthony Vazzana, Introduction to Number Theory Steven Furino, Ying Miao, and Jianxing Yin, Frames and Resolvable Designs: Uses, Constructions, and Existence Randy Goldberg and Lance Riek, A Practical Handbook of Speech Coders Jacob E. Goodman and Joseph O’Rourke, Handbook of Discrete and Computational Geometry, Second Edition Jonathan L. Gross, Combinatorial Methods with Computer Applications Jonathan L. Gross and Jay Yellen, Graph Theory and Its Applications, Second Edition Jonathan L. Gross and Jay Yellen, Handbook of Graph Theory Darrel R. Hankerson, Greg A. Harris, and Peter D. Johnson, Introduction to Information Theory and Data Compression, Second Edition Daryl D. Harms, Miroslav Kraetzl, Charles J. Colbourn, and John S. Devitt, Network Reliability: Experiments with a Symbolic Algebra Environment Leslie Hogben, Handbook of Linear Algebra Derek F. Holt with Bettina Eick and Eamonn A. O’Brien, Handbook of Computational Group Theory David M. Jackson and Terry I. Visentin, An Atlas of Smaller Maps in Orientable and Nonorientable Surfaces Richard E. Klima, Neil P. Sigmon, and Ernest L. Stitzinger, Applications of Abstract Algebra with Maple™ and MATLAB®, Second Edition
Continued Titles Patrick Knupp and Kambiz Salari, Verification of Computer Codes in Computational Science and Engineering William Kocay and Donald L. Kreher, Graphs, Algorithms, and Optimization Donald L. Kreher and Douglas R. Stinson, Combinatorial Algorithms: Generation Enumeration and Search Charles C. Lindner and Christopher A. Rodgers, Design Theory Hang T. Lau, A Java Library of Graph Algorithms and Optimization Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, Handbook of Applied Cryptography Richard A. Mollin, Algebraic Number Theory Richard A. Mollin, Codes: The Guide to Secrecy from Ancient to Modern Times Richard A. Mollin, Fundamental Number Theory with Applications, Second Edition Richard A. Mollin, An Introduction to Cryptography, Second Edition Richard A. Mollin, Quadratics Richard A. Mollin, RSA and Public-Key Cryptography Carlos J. Moreno and Samuel S. Wagstaff, Jr., Sums of Squares of Integers Dingyi Pei, Authentication Codes and Combinatorial Designs Kenneth H. Rosen, Handbook of Discrete and Combinatorial Mathematics Douglas R. Shier and K.T. Wallenius, Applied Mathematical Modeling: A Multidisciplinary Approach Jörn Steuding, Diophantine Analysis Douglas R. Stinson, Cryptography: Theory and Practice, Third Edition Roberto Togneri and Christopher J. deSilva, Fundamentals of Information Theory and Coding Design W. D. Wallis, Introduction to Combinatorial Designs, Second Edition Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography, Second Edition
DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN GARY CHARTRAND Western Michigan University Kalamazoo, MI, U.S.A. Ping Zhang Western Michigan University Kalamazoo, MI, U.S.A. Chromatic Graph Theory
Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-58488-800-0 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher can- not assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copy- right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro- vides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
PREFACE Beginning with the origin of the Four Color Problem in 1852, the field of graph colorings has developed into one of the most popular areas of graph theory. This book introduces graph theory with a coloring theme. It explores connections be- tween major topics in graph theory and graph colorings, including Ramsey numbers and domination, as well as such emerging topics as list colorings, rainbow colorings, distance colorings related to the Channel Assignment Problem, and vertex/edge dis- tinguishing colorings. Discussions of a historical, applied, and algorithmic nature are included. Each chapter in the text contains many exercises of varying levels of difficulty. There is also an appendix containing suggestions for study projects. The authors are honored to have been invited by CRC Press to write a textbook on graph colorings. With the enormous literature that exists on graph colorings and the dynamic nature of the subject, we were immediately faced with the challenge of determining which topics to include and, perhaps even more importantly, which topics to exclude. There are several instances when the same concept has been studied by different authors using different terminology and different notation. We were therefore required to make a number of decisions regarding terminology and notation. While nearly all mathematicians working on graph colorings use positive integers as colors, there are also some who use nonnegative integers. There are instances when colorings and labelings have been used synonymously. For the most part, colorings have been used when the primary goal has been either to minimize the number of colors or the largest color (equivalently, the span of colors). We decided that this book should be intended for one or more of the following purposes: • a course in graph theory with an emphasis on graph colorings, where this course could be either a beginning course in graph theory or a follow-up course to an elementary graph theory course, • a reading course on graph colorings, • a seminar on graph colorings, • as a reference book for individuals interested in graph colorings. To accomplish this, it has been our goal to write this book in an engaging, student- friendly style so that it contains carefully explained proofs and examples and con- tains many exercises of varying difficulty. This book consists of 15 chapters (Chapters 0-14). Chapter 0 provides some background on the origin of graph colorings – primarily giving a discussion of the Four Color Problem. For those readers who desire a more extensive discussion of the history and solution of the Four Color Problem, we recommend the interesting book by Robin Wilson, titled Four Colors Suffice: How the Map Problem Was Solved, published by Princeton University Press in 2002. To achieve the goal of having the book self-contained, Chapters 1-5 have been written to contain many of the fundamentals of graph theory that lie outside of vii
graph colorings. This includes basic terminology and results, trees and connec- tivity, Eulerian and Hamiltonian graphs, matchings and factorizations, and graph embeddings. The remainder of the book (Chapters 6-14) deal exclusively with graph colorings. Chapters 6 and 7 provide an introduction to vertex colorings and bounds for the chromatic number. The emphasis of Chapter 8 is vertex colorings of graphs embedded on surfaces. Chapter 9 discusses a variety of restricted ver- tex colorings, including list colorings. Chapter 10 introduces edge colorings, while Chapter 11 discusses monochromatic and rainbow edge colorings, including an in- troduction to Ramsey numbers. Chapter 11 also provides a discussion of the Road Coloring Problem. The main emphasis of Chapter 12 is complete vertex colorings. In Chapter 13, several distinguishing vertex and edge colorings are described. In Chapter 14 many distance-related vertex colorings are introduced, some inspired by the Channel Assignment Problem, as well as a discussion of domination in terms of vertex colorings. There is an Appendix listing fourteen topics for those who may be interested in pursuing some independent study. There are two sections containing references at the end of the book. The first of these, titled General References, contains a list of references, both for Chapter 0 and of a general nature for all succeeding chapters. The second such section (Bibliography) primarily contains a list of publications to which specific reference is made in the text. Finally, there is an Index of Names, listing individuals referred to in this book, an Index of Mathematical Terms, and a List of Symbols. There are many people we wish to thank. First, our thanks to mathemati- cians Ken Appel, Tiziana Calamoneri, Nicolaas de Bruijn, Ermelinda DeLaVi˜na, Stephen Locke, Staszek Radziszowski, Edward Schmeichel, Robin Thomas, Olivier Togni, and Avraham Trahtman for kindly providing us with information and com- municating with us on some topics. Thank you as well to our friends Shashi Kapoor and Al Polimeni for their interest and encouragement in this project. We especially want to thank Bob Stern, Executive Editor of CRC Press, Taylor & Francis Group, for his constant communication, encouragement, and interest and for suggesting this writing project to us. Finally, we thank Marsha Pronin, Project Coordinator, Samantha White, Editorial Assistant, and Jim McGovern, Project Editor for their cooperation. G.C. & P.Z. viii
Table of Contents 0. The Origin of Graph Colorings 1 1. Introduction to Graphs 27 1.1 Fundamental Terminology 27 1.2 Connected Graphs 30 1.3 Distance in Graphs 33 1.4 Isomorphic Graphs 37 1.5 Common Graphs and Graph Operations 39 1.6 Multigraphs and Digraphs 44 Exercises for Chapter 1 47 2. Trees and Connectivity 53 2.1 Cut-vertices, Bridges, and Blocks 53 2.2 Trees 56 2.3 Connectivity and Edge-Connectivity 59 2.4 Menger’s Theorem 63 Exercises for Chapter 2 67 3. Eulerian and Hamiltonian Graphs 71 3.1 Eulerian Graphs 71 3.2 de Bruijn Digraphs 76 3.3 Hamiltonian Graphs 79 Exercises for Chapter 3 87 4. Matchings and Factorization 91 4.1 Matchings 91 4.2 Independence in Graphs 98 4.3 Factors and Factorization 100 Exercises for Chapter 4 106 5. Graph Embeddings 109 5.1 Planar Graphs and the Euler Identity 109 5.2 Hamiltonian Planar Graphs 118 ix
5.3 Planarity Versus Nonplanarity 120 5.4 Embedding Graphs on Surfaces 131 5.5 The Graph Minor Theorem 139 Exercises for Chapter 5 141 6. Introduction to Vertex Colorings 147 6.1 The Chromatic Number of a Graph 147 6.2 Applications of Colorings 153 6.3 Perfect Graphs 160 Exercises for Chapter 6 170 7. Bounds for the Chromatic Number 175 7.1 Color-Critical Graphs 175 7.2 Upper Bounds and Greedy Colorings 179 7.3 Upper Bounds and Oriented Graphs 189 7.4 The Chromatic Number of Cartesian Products 195 Exercises for Chapter 7 200 8. Coloring Graphs on Surfaces 205 8.1 The Four Color Problem 205 8.2 The Conjectures of Haj´os and Hadwiger 208 8.3 Chromatic Polynomials 211 8.4 The Heawood Map-Coloring Problem 217 Exercises for Chapter 8 219 9. Restricted Vertex Colorings 223 9.1 Uniquely Colorable Graphs 223 9.2 List Colorings 230 9.3 Precoloring Extensions of Graphs 240 Exercises for Chapter 9 245 10. Edge Colorings of Graphs 249 10.1 The Chromatic Index and Vizing’s Theorem 249 10.2 Class One and Class Two Graphs 255 10.3 Tait Colorings 262 10.4 Nowhere-Zero Flows 269 10.5 List Edge Colorings 279 x
10.6 Total Colorings of Graphs 282 Exercises for Chapter 10 284 11. Monochromatic and Rainbow Colorings 289 11.1 Ramsey Numbers 289 11.2 Tur´an’s Theorem 296 11.3 Rainbow Ramsey Numbers 299 11.4 Rainbow Numbers of Graphs 306 11.5 Rainbow-Connected Graphs 314 11.6 The Road Coloring Problem 320 Exercises for Chapter 11 324 12. Complete Colorings 329 12.1 The Achromatic Number of a Graph 329 12.2 Graph Homomorphisms 335 12.3 The Grundy Number of a Graph 349 Exercises for Chapter 12 356 13. Distinguishing Colorings 359 13.1 Edge-Distinguishing Vertex Colorings 359 13.2 Vertex-Distinguishing Edge Colorings 370 13.3 Vertex-Distinguishing Vertex Colorings 379 13.4 Neighbor-Distinguishing Edge Colorings 385 Exercises for Chapter 13 391 14. Colorings, Distance, and Domination 397 14.1 T-Colorings 397 14.2 L(2, 1)-Colorings 403 14.3 Radio Colorings 410 14.4 Hamiltonian Colorings 417 14.5 Domination and Colorings 425 14.6 Epilogue 434 Exercises for Chapter 14 434 Appendix: Study Projects 439 General References 446 Bibliography 453 Index (Names and Mathematical Terms) 465 List of Symbols 480 xi
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To Frank Harary (1921–2005) for inspiring so many, including us, to look for the beauty in graph theory. xiii
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Chapter 0 The Origin of Graph Colorings If the countries in a map of South America (see Figure 1) were to be colored in such a way that every two countries with a common boundary are colored differently, then this map could be colored using only four colors. Is this true of every map? While it is not difficult to color a map of South America with four colors, it is not possible to color this map with less than four colors. In fact, every two of Brazil, Argentina, Bolivia, and Paraguay are neighboring countries and so four colors are required to color only these four countries. It is probably clear why we might want two countries colored differently if they have a common boundary – so they can be easily distinguished as different coun- tries in the map. It may not be clear, however, why we would think that four colors would be enough to color the countries of every map. After all, we can probably envision a complicated map having a large number of countries with some countries having several neighboring countries, so constructed that a great many colors might possibly be needed to color the entire map. Here we understand neighboring coun- tries to mean two countries with a boundary line in common, not simply a single point in common. While this problem may seem nothing more than a curiosity, it is precisely this problem that would prove to intrigue so many for so long and whose attempted solutions would contribute so significantly to the development of the area of math- ematics known as Graph Theory and especially to the subject of graph colorings: Chromatic Graph Theory. This map coloring problem would eventually acquire a name that would become known throughout the mathematical world. The Four Color Problem Can the countries of every map be colored with four or fewer colors so that every two countries with a common boundary are colored differently? 1
2 CHAPTER 0. THE ORIGIN OF GRAPH COLORINGS .................................................... ...................................... .............................................................. ...................................... ............................................................ ...................................... .............................................................. ...................................... ......................................................... ...................................... ............................................................................................................................................................................................................................................................................................................................ ............................................................... 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Venezuela Uruguay Falkland Islands Colombia Bolivia Chile Atlantic Ocean Brazil French Guiana Guyana Suriname Ecuador Peru Pacific Ocean Argentina Paraguay Figure 1: Map of South America Many of the concepts, theorems, and problems of Graph Theory lie in the shad- ows of the Four Color Problem. Indeed . . . Graph Theory is an area of mathematics whose past is always present. Since the maps we consider can be real or imagined, we can think of maps being divided into more general regions, rather than countries, states, provinces, or some other geographic entities. So just how did the Four Color Problem begin? It turns out that this question has a rather well-documented answer. On 23 October 1852, a student, namely Fred- erick Guthrie (1833–1886), at University College London visited his mathematics professor, the famous Augustus De Morgan (1806–1871), to describe an apparent mathematical discovery of his older brother Francis. While coloring the counties of a map of England, Francis Guthrie (1831–1899) observed that he could color them
3 with four colors, which led him to conjecture that no more than four colors would be needed to color the regions of any map. The Four Color Conjecture The regions of every map can be colored with four or fewer colors in such a way that every two regions sharing a common boundary are colored differently. Two years earlier, in 1850, Francis had earned a Bachelor of Arts degree from University College London and then a Bachelor of Laws degree in 1852. He would later become a mathematics professor himself at the University of Cape Town in South Africa. Francis developed a lifelong interest in botany and his extensive collection of flora from the Cape Peninsula would later be placed in the Guthrie Herbarium in the University of Cape Town Botany Department. Several rare species of flora are named for him. Francis Guthrie attempted to prove the Four Color Conjecture and although he thought he may have been successful, he was not completely satisfied with his proof. Francis discussed his discovery with Frederick. With Francis’ approval, Frederick mentioned the statement of this apparent theorem to Professor De Morgan, who expressed pleasure with it and believed it to be a new result. Evidently Frederick asked Professor De Morgan if he was aware of an argument that would establish the truth of the theorem. This led De Morgan to write a letter to his friend, the famous Irish mathemati- cian Sir William Rowan Hamilton (1805–1865) in Dublin. These two mathematical giants had corresponded for years, although apparently had met only once. De Mor- gan wrote (in part): My dear Hamilton: A student of mine asked me to day to give him a reason for a fact which I did not know was a fact – and do not yet. He says that if a figure be any how divided and the compartments differently coloured so that figures with any portion of common boundary lines are differently coloured – four colours may be wanted but not more – the following is his case in which four are wanted. ............................................................................................. ......................................................................................................................................................................... BA C DareA B C D names of colours Query cannot a necessity for five or more be invented . . . My pupil says he guessed it colouring a map of England . . .. The more I think of it the more evident it seems. If you retort with some very simple case which makes me out a stupid animal, I think I must do as the Sphynx did . . .
4 CHAPTER 0. THE ORIGIN OF GRAPH COLORINGS In De Morgan’s letter to Hamilton, he refers to the “Sphynx” (or Sphinx). While the Sphinx is a male statue of a lion with the head of a human in ancient Egypt which guards the entrance to a temple, the Greek Sphinx is a female creature of bad luck who sat atop a rock posing the following riddle to all those who pass by: What animal is that which in the morning goes on four feet, at noon on two, and in the evening upon three? Those who did not solve the riddle were killed. Only Oedipus (the title character in Oedipus Rex by Sophocles, a play about how people do not control their own destiny) answered the riddle correctly as “Man”, who in childhood (the morning of life) creeps on hands and knees, in manhood (the noon of life) walks upright, and in old age (the evening of life) walks with the aid of a cane. Upon learning that her riddle had been solved, the Sphinx cast herself from the rock and perished, a fate De Morgan had envisioned for himself if his riddle (the Four Color Problem) had an easy and immediate solution. In De Morgan’s letter to Hamilton, De Morgan attempted to explain why the problem appeared to be difficult. He followed this explanation by writing: But it is tricky work and I am not sure of all convolutions – What do you say? And has it, if true been noticed? Among Hamilton’s numerous mathematical accomplishments was his remark- able work with quaternions. Hamilton’s quaternions are a 4-dimensional system of numbers of the form a + bi + cj + dk, where a, b, c, d ∈ R and i2 = j2 = k2 = −1. When c = d = 0, these numbers are the 2-dimensional system of complex numbers; while when b = c = d = 0, these numbers are simply real numbers. Although it is commonplace for binary operations in algebraic structures to be commutative, such is not the case for products of quaternions. For example, i · j = k but j · i = −k. Since De Morgan had shown an interest in Hamilton’s research on quaternions as well as other subjects Hamilton had studied, it is likely that De Morgan expected an enthusiastic reply to his letter to Hamilton. Such was not the case, however. Indeed, on 26 October 1852, Hamilton gave a quick but unexpected response: I am not likely to attempt your “quaternion” of colours very soon. Hamilton’s response did nothing however to diminish De Morgan’s interest in the Four Color Problem. Since De Morgan’s letter to Hamilton did not mention Frederick Guthrie by name, there may be reason to question whether Frederick was in fact the student to whom De Morgan was referring and that it was Frederick’s older brother Francis who was the originator of the Four Color Problem. In 1852 Frederick Guthrie was a teenager. He would go on to become a distin- guished physics professor and founder of the Physical Society in London. An area that he studied is the science of thermionic emission – first reported by Frederick Guthrie in 1873. He discovered that a red-hot iron sphere with a positive charge
5 would lose its charge. This effect was rediscovered by the famous American inventor Thomas Edison early in 1880. It was during 1880 (only six years before Frederick died) that Frederick wrote: Some thirty years ago, when I was attending Professor De Morgan’s class, my brother, Francis Guthrie, who had recently ceased to attend then (and who is now professor of mathematics at the South African University, Cape Town), showed me the fact that the greatest necessary number of colours to be used in colouring a map so as to avoid identity colour in lineally contiguous districts is four. I should not be justified, after this lapse of time, in trying to give his proof, but the critical dia- gram was as in the margin. ........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... .............................................. ................................................. 1 4 32 With my brother’s permission I submitted the theorem to Professor De Morgan, who expressed himself very pleased with it; accepted it as new; and, as I am informed by those who subsequently attended his classes, was in the habit of acknowledging where he had got his infor- mation. If I remember rightly, the proof which my brother gave did not seem altogether satisfactory to himself; but I must refer to him those interested in the subject. . . .. The first statement in print of the Four Color Problem evidently occurred in an anonymous review written in the 14 April 1860 issue of the literary journal Athenaeum. Although the author of the review was not identified, De Morgan was quite clearly the writer. This review led to the Four Color Problem becoming known in the United States. The Four Color Problem came to the attention of the American mathematician Charles Sanders Peirce (1839–1914), who found an example of a map drawn on a torus (a donut-shaped surface) that required six colors. (As we will see in Chapter 5, there is an example of a map drawn on a torus that requires seven colors.) Peirce expressed great interest in the Four Color Problem. In fact, he visited De Morgan in 1870, who by that time was experiencing poor health. Indeed, De Morgan died the following year. Not only had De Morgan made little progress towards a solution of the Four Color Problem at the time of his death, overall interest in this problem had faded. While Peirce continued to attempt to solve the problem, De Morgan’s British acquaintances appeared to pay little attention to the problem – with at least one notable exception.
6 CHAPTER 0. THE ORIGIN OF GRAPH COLORINGS Arthur Cayley (1821–1895) graduated from Trinity College, Cambridge in 1842 and then received a fellowship from Cambridge, where he taught for four years. Afterwards, because of the limitations on his fellowship, he was required to choose a profession. He chose law, but only as a means to make money while he could continue to do mathematics. During 1849–1863, Cayley was a successful lawyer but published some 250 research papers during this period, including many for which he is well known. One of these was his pioneering paper on matrix algebra. Cayley was famous for his work on algebra, much of which was done with the British mathematician James Joseph Sylvester (1814–1897), a former student of De Morgan. In 1863 Cayley was appointed a professor of mathematics at Cambridge. Two years later, the London Mathematical Society was founded at University College London and would serve as a model for the American Mathematical Society, founded in 1888. De Morgan became the first president of the London Mathematical Society, followed by Sylvester and then Cayley. During a meeting of the Society on 13 June 1878, Cayley raised a question about the Four Color Problem that brought renewed attention to the problem: Has a solution been given of the statement that in colouring a map of a country, divided into counties, only four distinct colours are required, so that no two adjacent counties should be painted in the same colour? This question appeared in the Proceedings of the Society’s meeting. In the April 1879 issue of the Proceedings of the Royal Geographical Society, Cayley reported: I have not succeeded in obtaining a general proof; and it is worth while to explain wherein the difficulty consists. Cayley observed that if a map with a certain number of regions has been colored with four colors and a new map is obtained by adding a new region, then there is no guarantee that the new map can be colored with four colors – without first recoloring the original map. This showed that any attempted proof of the Four Color Conjecture using a proof by mathematical induction would not be straightforward. Another possible proof technique to try would be proof by contradiction. Applying this technique, we would assume that the Four Color Conjecture is false. This would mean that there are some maps that cannot be colored with four colors. Among the maps that require five or more colors are those with a smallest number of regions. Any one of these maps constitutes a minimum counterexample. If it could be shown that no minimum counterexample could exist, then this would establish the truth of the Four Color Conjecture. For example, no minimum counterexample M could possibly contain a region R surrounded by three regions R1, R2, and R3 as shown in Figure 2(a). In this case, we could shrink the region R to a point, producing a new map M′ with one less region. The map M′ can then be colored with four colors, only three of which are used to color R1, R2, and R3 as in Figure 2(b). Returning to the original map M, we see that there is now an available color for R as shown in Figure 2(c),
7 implying that M could be colored with four colors after all, thereby producing a contradiction. Certainly, if the map M contains a region surrounded by fewer than three regions, a contradiction can be obtained in the same manner. ...................................................................................................................................................................................................................................... ....................................................................................................................................................................................................................................... .................................................................................. ..................................................................................... .................................................................................. .................................................................................. ..................................................................................... ...................................... ...................................... ..................................................................................... R (a) (c) blue green red greenR3 (b) in M′ in Min M yellow blueR2 R1 yellow Figure 2: A region surrounded by three regions in a map Suppose, however, that the map M contained no region surrounded by three or fewer regions but did contain a region R surrounded by four regions, say R1, R2, R3, R4, as shown in Figure 3(a). If, once again, we shrink the region R to a point, producing a map M′ with one less region, then we know that M′ can be colored with four colors. If two or three colors are used to color R1, R2, R3, R4, then we can return to M and there is a color available for R. However, this technique does not work if the regions R1, R2, R3, R4 are colored with four distinct colors, as shown in Figure 3(b). ...................................................................................................................................................................................................................................... ...................................... .................................................................................................................................................................................... R R3 R1 green red R4 blue yellow R2 in M(a) (b) in M′ Figure 3: A region surrounded by four regions in a map What we can do in this case, however, is to determine whether the map M′ has a chain of regions, beginning at R1 and ending at R3, all of which are colored red or green. If no such chain exists, then the two colors of every red-green chain of regions beginning at R1 can be interchanged. We can then return to the map M, where the color red is now available for R. That is, the map M can be colored with four colors, producing a contradiction. But what if a red-green chain of regions beginning at R1 and ending at R3 exists? (See Figure 4, where r, b, g, y denote the colors red, blue, green, yellow.) Then interchanging the colors red and green offers no benefit to us. However, in this case, there can be no blue-yellow chain of regions, beginning at R2 and ending at R4. Then the colors of every blue-yellow chain of
8 CHAPTER 0. THE ORIGIN OF GRAPH COLORINGS regions beginning at R2 can be interchanged. Returning to M, we see that the color blue is now available for R, which once again says that M can be colored with four colors and produces a contradiction. .................... .......................... g rg r g g r R3 R1 R4 R2 b yg R r r ..................................................................................................................................................................................................................................................................................................................... ............................................................................................................................................................................................................................................................................................................... ......................... .............................................................................. ................................................................................................................................................................................................... .............................................................................................................................................................................................................................................................. ............. ..................................................................................................................................................................................................... ............................................................................................................................................................................................................................................................ ....................................................................................................................................................................................................................................................................................................................................................................... .......... .................................................................................................................................................................................................................................................. ................................... .................................................................................................................................................................................................................................................................................................................................................. ............................................................................................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................................................................... ................................................................................................................................................................................................... ......................................................................................................... ....................................................................... ........................................................................................................................................................................ Figure 4: A a red-green chain of regions from R1 to R3 It is possible to show (as we will see in Chapter 5) that a map may contain no region that is surrounded by four or fewer neighboring regions. Should this occur however, such a map must contain a region surrounded by exactly five neighboring regions. We mentioned that James Joseph Sylvester worked with Arthur Cayley and served as the second president of the London Mathematical Society. Sylvester, a superb mathematician himself, was invited to join the mathematics faculty of the newly founded Johns Hopkins University in Baltimore, Maryland in 1875. Included among his attempts to inspire more research at the university was his founding in 1878 of the American Journal of Mathematics, of which he held the position of editor-in-chief. While the goal of the journal was to serve American mathematicians, foreign submissions were encouraged as well, including articles from Sylvester’s friend Cayley. Among those who studied under Arthur Cayley was Alfred Bray Kempe (1849– 1922). Despite his great enthusiasm for mathematics, Kempe took up a career in the legal profession. Kempe was present at the meeting of the London Mathematical Society in which Cayley had inquired about the status of the Four Color Problem. Kempe worked on the problem and obtained a solution in 1879. Indeed, on 17 July 1879 a statement of Kempe’s accomplishment appeared in the British journal Nature, with the complete proof published in Volume 2 of Sylvester’s American Journal of Mathematics. Kempe’s approach for solving the Four Color Problem essentially followed the technique described earlier. His technique involved locating a region R in a map M such that R is surrounded by five or fewer neighboring regions and showing that for every coloring of M (minus the region R) with four colors, there is a coloring of the entire map M with four colors. Such an argument would show that M could not be a minimum counterexample. We saw how such a proof would proceed if R were
9 surrounded by four or fewer neighboring regions. This included looking for chains of regions whose colors alternate between two colors and then interchanging these colors, if appropriate, to arrive at a coloring of the regions of M (minus R) with four colors so that the neighboring regions of R used at most three of these colors and thereby leaving a color available for R. In fact, these chains of regions became known as Kempe chains, for it was Kempe who originated this idea. There was one case, however, that still needed to be resolved, namely the case where no region in the map was surrounded by four or fewer neighboring regions. As we noted, the map must then contain some region R surrounded by exactly five neighboring regions. At least three of the four colors must be used to color the five neighboring regions of R. If only three colors are used to color these five regions, then a color is available for R. Hence we are left with the single situation in which all four colors are used to color the five neighboring regions surrounding R (see Figure 5), where once again r, b, g, y indicate the colors red, blue, green, yellow. ........................................................... ...................................... ........................................................................................................... ............................................................ ...................................... .................................................................... ...................................... ............................................................... ...................................... .............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ...................................................... ............................................................. ...................................................................... ....................................................................... ........................................................................................ ............................................................................................................................................................................................................................................................................................................................................................................................................................................................... ...................................................................................................................................................................................................................................................................................... .......................................................................................................................................................................................................................................... ..................................................................................................................................................................................................................................... R g y b r b R4 R2 R5 R3 R1 Figure 5: The final case in Kempe’s solution of the Four Color Problem Let’s see how Kempe handled this final case. Among the regions adjacent to R, only the region R1 is colored yellow. Consider all the regions of the map M that are colored either yellow or red and that, beginning at R1, can be reached by an alternating sequence of neighboring yellow and red regions, that is, by a yellow-red Kempe chain. If the region R3 (which is the neighboring region of R colored red) cannot be reached by a yellow-red Kempe chain, then the colors yellow and red can be interchanged for all regions in M that can be reached by a yellow-red Kempe chain beginning at R1. This results in a coloring of all regions in M (except R) in which neighboring regions are colored differently and such that each neighboring region of R is colored red, blue, or green. We can then color R yellow to arrive at a 4-coloring of the entire map M. From this, we may assume that the region R3 can be reached by a yellow-red Kempe chain beginning at R1. (See Figure 6.) Let’s now look at the region R5, which is colored green. We consider all regions of M colored green or red and that, beginning at R5, can be reached by a green-red Kempe chain. If the region R3 cannot be reached by a green-red Kempe chain that begins at R5, then the colors green and red can be interchanged for all regions in M
10 CHAPTER 0. THE ORIGIN OF GRAPH COLORINGS .............................................................. ...................................... ............................................................. ...................................... .................................................................. ...................................... .............................................................. ...................................... ..................................................................................................... .............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ...................................................... ............................................................. ...................................................................... ....................................................................... ........................................................................................ ............................................................................................................................................................................................................................................................................................................................................................................................................................................................... ...................................................................................................................................................................................................................................................................................... ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ..................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................................................................................................................................................................................................................. ....................................................................................................................................................................................................................................................................................................................................... ............................................................................................................................................................................................................................................................................................... ............................................................................................ ............................................................................................................................................................................................................................... R r r y y y r y r b g b R3 R4 R5 R1 R2 Figure 6: A yellow-red Kempe chain in the map M that can be reached by a green-red Kempe chain beginning at R5. Upon doing this, a 4-coloring of all regions in M (except R) is obtained, in which each neighboring region of R is colored red, blue, or yellow. We can then color R green to produce a 4-coloring of the entire map M. We may therefore assume that R3 can be reached by a green-red Kempe chain that begins at R5. (See Figure 7.) .............................................................................................. ................................................... ...................................... .......................................................................... ...................................... ............................................................. ...................................... ............................................................................................................... ...................................... 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............................................................................................................................................................. y y g r r r r r y r g r g g b b gy RR2 r R4 R3 R5 R1 Figure 7: Yellow-red and green-red Kempe chains in the map M Because there is a ring of regions consisting of R and a green-red Kempe chain, there cannot be a blue-yellow Kempe chain in M beginning at R4 and ending at R1. In addition, because there is a ring of regions consisting of R and a yellow-red Kempe chain, there is no blue-green Kempe chain in M beginning at R2 and ending at R5. Hence we interchange the colors blue and yellow for all regions in M that can be reached by a blue-yellow Kempe chain beginning at R4 and interchange the colors
11 blue and green for all regions in M that can be reached by a blue-green Kempe chain beginning at R2. Once these two color interchanges have been performed, each of the five neighboring regions of R is colored red, yellow, or green. Then R can be colored blue and a 4-coloring of the map M has been obtained, completing the proof. As it turned out, the proof given by Kempe contained a fatal flaw, but one that would go unnoticed for a decade. Despite the fact that Kempe’s attempted proof of the Four Color Problem was erroneous, he made a number of interesting observations in his article. He noticed that if a piece of tracing paper was placed over a map and a point was marked on the tracing paper over each region of the map and two points were joined by a line segment whenever the corresponding regions had a common boundary, then a diagram of a “linkage” was produced. Furthermore, the problem of determining whether the regions of the map can be colored with four colors so that neighboring regions are colored differently is the same problem as determining whether the points in the linkage can be colored with four colors so that every two points joined by a line segment are colored differently. (See Figure 8.) ................................................................................................................................................................................................................................................................................................................................. .......................................................................................................................................................................................................................... ............................................................................................................................................................................................. ............................................................................................................................................................................................. 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ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp s pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppps s s s s s s s s Figure 8: A map and corresponding planar graph In 1878 Sylvester referred to a linkage as a graph and it is this terminology that became accepted. Later it became commonplace to refer to the points and lines of a linkage as the vertices and edges of the graph (with “vertex” being the singular of “vertices”). Since the graphs constructed from maps in this manner (referred to as the dual graph of the map) can themselves be drawn in the plane without two edges (line segments) intersecting, these graphs were called planar graphs. A planar graph that is actually drawn in the plane without any of its edges intersecting is called a plane graph. In terms of graphs, the Four Color Conjecture could then be restated. The Four Color Conjecture The vertices of every planar graph can be colored with four or fewer colors in such a way that every two vertices joined by an edge are colored differently. Indeed, the vast majority of this book will be devoted to coloring graphs (not coloring maps) and, in fact, to coloring graphs in general, not only planar graphs.
12 CHAPTER 0. THE ORIGIN OF GRAPH COLORINGS The colouring of abstract graphs is a generalization of the colouring of maps, and the study of the colouring of abstract graphs . . . opens a new chapter in the combinatorial part of mathematics. Gabriel Andrew Dirac (1951) For the present, however, we continue our discussion in terms of coloring the regions of maps. Kempe’s proof of the theorem, which had become known as the Four Color Theorem, was accepted both within the United States and England. Arthur Cayley had accepted Kempe’s argument as a valid proof. This led to Kempe being elected as a Fellow of the Royal Society in 1881. The Four Color Theorem The regions of every map can be colored with four or fewer colors so that every two adjacent regions are colored differently. Among the many individuals who had become interested in the Four Color Problem was Charles Lutwidge Dodgson (1832–1898), an Englishman with a keen interest in mathematics and puzzles. Dodgson was better known, however, under his pen-name Lewis Carroll and for his well-known books Alice’s Adventures in Wonderland and Through the Looking-Glass and What Alice Found There. Another well-known individual with mathematical interests, but whose primary occupation was not that of a mathematician, was Frederick Temple (1821–1902), Bishop of London and who would later become the Archbishop of Canterbury. Like Dodgson and others, Temple had a fondness for puzzles. Temple showed that it was impossible to have five mutually neighboring regions in any map and from this concluded that no map required five colors. Although Temple was correct about the non-existence of five mutually neighboring regions in a map, his conclusion that this provided a proof of the Four Color Conjecture was incorrect. There was historical precedence about the non-existence of five mutually ad- jacent regions in any map. In 1840 the famous German mathematician August M¨obius (1790–1868) reportedly stated the following problem, which was proposed to him by the philologist Benjamin Weiske (1748–1809). Problem of Five Princes There was once a king with five sons. In his will, he stated that after his death his kingdom should be divided into five regions in such a way that each region should have a common boundary with the other four. Can the terms of the will be satisfied? As we noted, the conditions of the king’s will cannot be met. This problem illustrates M¨obius’ interest in topology, a subject of which M¨obius was one of the early pioneers. In a memoir written by M¨obius and only discovered after his death, he discussed properties of one-sided surfaces, which became known as M¨obius strips (even though it was determined that Johann Listing (1808–1882) had discovered these earlier).
13 In 1885 the German geometer Richard Baltzer (1818–1887) also lectured on the non-existence of five mutually adjacent regions. In the published version of his lecture, it was incorrectly stated that the Four Color Theorem followed from this. This error was repeated by other writers until the famous geometer Harold Scott MacDonald Coxeter (1907–2003) corrected the matter in 1959. Mistakes concerning the Four Color Problem were not limited to mathematical errors however. Prior to establishing Francis Guthrie as the true and sole originator of the Four Color Problem, it was often stated in print that cartographers were aware that the regions of every map could be colored with four or less colors so that adjacent regions are colored differently. The well-known mathematical historian Kenneth O. May (1915–1977) investigated this claim and found no justification to it. He conducted a study of atlases in the Library of Congress and found no evidence of attempts to minimize the number of colors used in maps. Most maps used more than four colors and even when four colors were used, often less colors could have been used. There was never a mention of a “four color theorem”. Another mathematician of note around 1880 was Peter Guthrie Tait (1831– 1901). In addition to being a scholar, he was a golf enthusiast. His son Frederick Guthrie Tait was a champion golfer and considered a national hero in Scotland. The first golf biography ever written was about Frederick Tait. Indeed, the Freddie Tait Golf Week is held every year in Kimberley, South Africa to commemorate his life as a golfer and soldier. He was killed during the Anglo-Boer War of 1899–1902. Peter Guthrie Tait had heard of the Four Color Conjecture through Arthur Cayley and was aware of Kempe’s solution. He felt that Kempe’s solution of the Four Color Problem was overly long and gave several shorter solutions of the problem, all of which turned out to be incorrect. Despite this, one of his attempted proofs contained an interesting and useful idea. A type of map that is often encountered is a cubic map, in which there are exactly three boundary lines at each meeting point. In fact, every map M that has no region completely surrounded by another region can be converted into a cubic map M′ by drawing a circle about each meeting point in M′ and creating new meeting points and one new region (see Figure 9). If the map M′ can be colored with four colors, then so can M. .............................................................................................................................................................................................................................................................................................................................................................................. ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ ...................................... .............................................................................................................................................................................................. ...................................................................................................................................................................................................... .................................................................................................. ...................................... ............................................. ...................................... .......................................... ........................................ .................................................................................................... in M in M′ Figure 9: Converting a map into a cubic map Tait’s idea was to consider coloring the boundary lines of cubic maps. In fact, he stated as a lemma that:
14 CHAPTER 0. THE ORIGIN OF GRAPH COLORINGS The boundary lines of every cubic map can always be colored with three colors so that the three lines at each meeting point are colored differently. Tait also mentioned that this lemma could be easily proved and showed how the lemma could be used to prove the Four Color Theorem. Although Tait was correct that this lemma could be used to to prove the Four Color Theorem, he was incorrect when he said that the lemma could be easily proved. Indeed, as it turned out, this lemma is equivalent to the Four Color Theorem and, of course, is equally difficult to prove. (We will discuss Tait’s coloring of the boundary lines of cubic maps in Chapter 10.) The next important figure in the history of the Four Color Problem was Percy John Heawood (1861–1955), who spent the period 1887–1939 as a lecturer, professor, and vice-chancellor at Durham College in England. When Heawood was a student at Oxford University in 1880, one of his teachers was Professor Henry Smith who spoke often of the Four Color Problem. Heawood read Kempe’s paper and it was he who discovered the serious error in the proof. In 1889 Heawood wrote a paper of his own, published in 1890, in which he presented the map shown in Figure 10. r g b y y b ........................................................................................................ ........................................................................................................................................................................................................................................................................................................................................................................................................................................................... ............................................................................ ................................................................................................. ........................................................................................................ 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Figure 10: Heawood’s counterexample to Kempe’s proof In the Heawood map, two of the five neighboring regions surrounding the uncol- ored region R are colored red; while for each of the colors blue, yellow, and green, there is exactly one neighboring region of R with that color. According to Kempe’s argument, since blue is the color of the region that shares a boundary with R as well as with the two neighboring regions of R colored red, we are concerned with whether this map contains a blue-yellow Kempe chain between two neighboring re- gions of R as well as a blue-green Kempe chain between two neighboring regions of
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Chromatic graph theory

  • 1.
  • 3. DISCRETE MATHEMATICS ITS APPLICATIONS Series Editor Kenneth H. Rosen, Ph.D. Juergen Bierbrauer, Introduction to Coding Theory Francine Blanchet-Sadri, Algorithmic Combinatorics on Partial Words Richard A. Brualdi and Drago˘s Cvetkovi´c, A Combinatorial Approach to Matrix Theory and Its Applications Kun-Mao Chao and Bang Ye Wu, Spanning Trees and Optimization Problems Charalambos A. Charalambides, Enumerative Combinatorics Gary Chartrand and Ping Zhang, Chromatic Graph Theory Henri Cohen, Gerhard Frey, et al., Handbook of Elliptic and Hyperelliptic Curve Cryptography Charles J. Colbourn and Jeffrey H. Dinitz, Handbook of Combinatorial Designs, Second Edition Martin Erickson and Anthony Vazzana, Introduction to Number Theory Steven Furino, Ying Miao, and Jianxing Yin, Frames and Resolvable Designs: Uses, Constructions, and Existence Randy Goldberg and Lance Riek, A Practical Handbook of Speech Coders Jacob E. Goodman and Joseph O’Rourke, Handbook of Discrete and Computational Geometry, Second Edition Jonathan L. Gross, Combinatorial Methods with Computer Applications Jonathan L. Gross and Jay Yellen, Graph Theory and Its Applications, Second Edition Jonathan L. Gross and Jay Yellen, Handbook of Graph Theory Darrel R. Hankerson, Greg A. Harris, and Peter D. Johnson, Introduction to Information Theory and Data Compression, Second Edition Daryl D. Harms, Miroslav Kraetzl, Charles J. Colbourn, and John S. Devitt, Network Reliability: Experiments with a Symbolic Algebra Environment Leslie Hogben, Handbook of Linear Algebra Derek F. Holt with Bettina Eick and Eamonn A. O’Brien, Handbook of Computational Group Theory David M. Jackson and Terry I. Visentin, An Atlas of Smaller Maps in Orientable and Nonorientable Surfaces Richard E. Klima, Neil P. Sigmon, and Ernest L. Stitzinger, Applications of Abstract Algebra with Maple™ and MATLAB®, Second Edition
  • 4. Continued Titles Patrick Knupp and Kambiz Salari, Verification of Computer Codes in Computational Science and Engineering William Kocay and Donald L. Kreher, Graphs, Algorithms, and Optimization Donald L. Kreher and Douglas R. Stinson, Combinatorial Algorithms: Generation Enumeration and Search Charles C. Lindner and Christopher A. Rodgers, Design Theory Hang T. Lau, A Java Library of Graph Algorithms and Optimization Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, Handbook of Applied Cryptography Richard A. Mollin, Algebraic Number Theory Richard A. Mollin, Codes: The Guide to Secrecy from Ancient to Modern Times Richard A. Mollin, Fundamental Number Theory with Applications, Second Edition Richard A. Mollin, An Introduction to Cryptography, Second Edition Richard A. Mollin, Quadratics Richard A. Mollin, RSA and Public-Key Cryptography Carlos J. Moreno and Samuel S. Wagstaff, Jr., Sums of Squares of Integers Dingyi Pei, Authentication Codes and Combinatorial Designs Kenneth H. Rosen, Handbook of Discrete and Combinatorial Mathematics Douglas R. Shier and K.T. Wallenius, Applied Mathematical Modeling: A Multidisciplinary Approach Jörn Steuding, Diophantine Analysis Douglas R. Stinson, Cryptography: Theory and Practice, Third Edition Roberto Togneri and Christopher J. deSilva, Fundamentals of Information Theory and Coding Design W. D. Wallis, Introduction to Combinatorial Designs, Second Edition Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography, Second Edition
  • 5.
  • 6. DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN GARY CHARTRAND Western Michigan University Kalamazoo, MI, U.S.A. Ping Zhang Western Michigan University Kalamazoo, MI, U.S.A. Chromatic Graph Theory
  • 7. Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-58488-800-0 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher can- not assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copy- right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro- vides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
  • 8. PREFACE Beginning with the origin of the Four Color Problem in 1852, the field of graph colorings has developed into one of the most popular areas of graph theory. This book introduces graph theory with a coloring theme. It explores connections be- tween major topics in graph theory and graph colorings, including Ramsey numbers and domination, as well as such emerging topics as list colorings, rainbow colorings, distance colorings related to the Channel Assignment Problem, and vertex/edge dis- tinguishing colorings. Discussions of a historical, applied, and algorithmic nature are included. Each chapter in the text contains many exercises of varying levels of difficulty. There is also an appendix containing suggestions for study projects. The authors are honored to have been invited by CRC Press to write a textbook on graph colorings. With the enormous literature that exists on graph colorings and the dynamic nature of the subject, we were immediately faced with the challenge of determining which topics to include and, perhaps even more importantly, which topics to exclude. There are several instances when the same concept has been studied by different authors using different terminology and different notation. We were therefore required to make a number of decisions regarding terminology and notation. While nearly all mathematicians working on graph colorings use positive integers as colors, there are also some who use nonnegative integers. There are instances when colorings and labelings have been used synonymously. For the most part, colorings have been used when the primary goal has been either to minimize the number of colors or the largest color (equivalently, the span of colors). We decided that this book should be intended for one or more of the following purposes: • a course in graph theory with an emphasis on graph colorings, where this course could be either a beginning course in graph theory or a follow-up course to an elementary graph theory course, • a reading course on graph colorings, • a seminar on graph colorings, • as a reference book for individuals interested in graph colorings. To accomplish this, it has been our goal to write this book in an engaging, student- friendly style so that it contains carefully explained proofs and examples and con- tains many exercises of varying difficulty. This book consists of 15 chapters (Chapters 0-14). Chapter 0 provides some background on the origin of graph colorings – primarily giving a discussion of the Four Color Problem. For those readers who desire a more extensive discussion of the history and solution of the Four Color Problem, we recommend the interesting book by Robin Wilson, titled Four Colors Suffice: How the Map Problem Was Solved, published by Princeton University Press in 2002. To achieve the goal of having the book self-contained, Chapters 1-5 have been written to contain many of the fundamentals of graph theory that lie outside of vii
  • 9. graph colorings. This includes basic terminology and results, trees and connec- tivity, Eulerian and Hamiltonian graphs, matchings and factorizations, and graph embeddings. The remainder of the book (Chapters 6-14) deal exclusively with graph colorings. Chapters 6 and 7 provide an introduction to vertex colorings and bounds for the chromatic number. The emphasis of Chapter 8 is vertex colorings of graphs embedded on surfaces. Chapter 9 discusses a variety of restricted ver- tex colorings, including list colorings. Chapter 10 introduces edge colorings, while Chapter 11 discusses monochromatic and rainbow edge colorings, including an in- troduction to Ramsey numbers. Chapter 11 also provides a discussion of the Road Coloring Problem. The main emphasis of Chapter 12 is complete vertex colorings. In Chapter 13, several distinguishing vertex and edge colorings are described. In Chapter 14 many distance-related vertex colorings are introduced, some inspired by the Channel Assignment Problem, as well as a discussion of domination in terms of vertex colorings. There is an Appendix listing fourteen topics for those who may be interested in pursuing some independent study. There are two sections containing references at the end of the book. The first of these, titled General References, contains a list of references, both for Chapter 0 and of a general nature for all succeeding chapters. The second such section (Bibliography) primarily contains a list of publications to which specific reference is made in the text. Finally, there is an Index of Names, listing individuals referred to in this book, an Index of Mathematical Terms, and a List of Symbols. There are many people we wish to thank. First, our thanks to mathemati- cians Ken Appel, Tiziana Calamoneri, Nicolaas de Bruijn, Ermelinda DeLaVi˜na, Stephen Locke, Staszek Radziszowski, Edward Schmeichel, Robin Thomas, Olivier Togni, and Avraham Trahtman for kindly providing us with information and com- municating with us on some topics. Thank you as well to our friends Shashi Kapoor and Al Polimeni for their interest and encouragement in this project. We especially want to thank Bob Stern, Executive Editor of CRC Press, Taylor & Francis Group, for his constant communication, encouragement, and interest and for suggesting this writing project to us. Finally, we thank Marsha Pronin, Project Coordinator, Samantha White, Editorial Assistant, and Jim McGovern, Project Editor for their cooperation. G.C. & P.Z. viii
  • 10. Table of Contents 0. The Origin of Graph Colorings 1 1. Introduction to Graphs 27 1.1 Fundamental Terminology 27 1.2 Connected Graphs 30 1.3 Distance in Graphs 33 1.4 Isomorphic Graphs 37 1.5 Common Graphs and Graph Operations 39 1.6 Multigraphs and Digraphs 44 Exercises for Chapter 1 47 2. Trees and Connectivity 53 2.1 Cut-vertices, Bridges, and Blocks 53 2.2 Trees 56 2.3 Connectivity and Edge-Connectivity 59 2.4 Menger’s Theorem 63 Exercises for Chapter 2 67 3. Eulerian and Hamiltonian Graphs 71 3.1 Eulerian Graphs 71 3.2 de Bruijn Digraphs 76 3.3 Hamiltonian Graphs 79 Exercises for Chapter 3 87 4. Matchings and Factorization 91 4.1 Matchings 91 4.2 Independence in Graphs 98 4.3 Factors and Factorization 100 Exercises for Chapter 4 106 5. Graph Embeddings 109 5.1 Planar Graphs and the Euler Identity 109 5.2 Hamiltonian Planar Graphs 118 ix
  • 11. 5.3 Planarity Versus Nonplanarity 120 5.4 Embedding Graphs on Surfaces 131 5.5 The Graph Minor Theorem 139 Exercises for Chapter 5 141 6. Introduction to Vertex Colorings 147 6.1 The Chromatic Number of a Graph 147 6.2 Applications of Colorings 153 6.3 Perfect Graphs 160 Exercises for Chapter 6 170 7. Bounds for the Chromatic Number 175 7.1 Color-Critical Graphs 175 7.2 Upper Bounds and Greedy Colorings 179 7.3 Upper Bounds and Oriented Graphs 189 7.4 The Chromatic Number of Cartesian Products 195 Exercises for Chapter 7 200 8. Coloring Graphs on Surfaces 205 8.1 The Four Color Problem 205 8.2 The Conjectures of Haj´os and Hadwiger 208 8.3 Chromatic Polynomials 211 8.4 The Heawood Map-Coloring Problem 217 Exercises for Chapter 8 219 9. Restricted Vertex Colorings 223 9.1 Uniquely Colorable Graphs 223 9.2 List Colorings 230 9.3 Precoloring Extensions of Graphs 240 Exercises for Chapter 9 245 10. Edge Colorings of Graphs 249 10.1 The Chromatic Index and Vizing’s Theorem 249 10.2 Class One and Class Two Graphs 255 10.3 Tait Colorings 262 10.4 Nowhere-Zero Flows 269 10.5 List Edge Colorings 279 x
  • 12. 10.6 Total Colorings of Graphs 282 Exercises for Chapter 10 284 11. Monochromatic and Rainbow Colorings 289 11.1 Ramsey Numbers 289 11.2 Tur´an’s Theorem 296 11.3 Rainbow Ramsey Numbers 299 11.4 Rainbow Numbers of Graphs 306 11.5 Rainbow-Connected Graphs 314 11.6 The Road Coloring Problem 320 Exercises for Chapter 11 324 12. Complete Colorings 329 12.1 The Achromatic Number of a Graph 329 12.2 Graph Homomorphisms 335 12.3 The Grundy Number of a Graph 349 Exercises for Chapter 12 356 13. Distinguishing Colorings 359 13.1 Edge-Distinguishing Vertex Colorings 359 13.2 Vertex-Distinguishing Edge Colorings 370 13.3 Vertex-Distinguishing Vertex Colorings 379 13.4 Neighbor-Distinguishing Edge Colorings 385 Exercises for Chapter 13 391 14. Colorings, Distance, and Domination 397 14.1 T-Colorings 397 14.2 L(2, 1)-Colorings 403 14.3 Radio Colorings 410 14.4 Hamiltonian Colorings 417 14.5 Domination and Colorings 425 14.6 Epilogue 434 Exercises for Chapter 14 434 Appendix: Study Projects 439 General References 446 Bibliography 453 Index (Names and Mathematical Terms) 465 List of Symbols 480 xi
  • 13. xii
  • 14. To Frank Harary (1921–2005) for inspiring so many, including us, to look for the beauty in graph theory. xiii
  • 15. xiv
  • 16. Chapter 0 The Origin of Graph Colorings If the countries in a map of South America (see Figure 1) were to be colored in such a way that every two countries with a common boundary are colored differently, then this map could be colored using only four colors. Is this true of every map? While it is not difficult to color a map of South America with four colors, it is not possible to color this map with less than four colors. In fact, every two of Brazil, Argentina, Bolivia, and Paraguay are neighboring countries and so four colors are required to color only these four countries. It is probably clear why we might want two countries colored differently if they have a common boundary – so they can be easily distinguished as different coun- tries in the map. It may not be clear, however, why we would think that four colors would be enough to color the countries of every map. After all, we can probably envision a complicated map having a large number of countries with some countries having several neighboring countries, so constructed that a great many colors might possibly be needed to color the entire map. Here we understand neighboring coun- tries to mean two countries with a boundary line in common, not simply a single point in common. While this problem may seem nothing more than a curiosity, it is precisely this problem that would prove to intrigue so many for so long and whose attempted solutions would contribute so significantly to the development of the area of math- ematics known as Graph Theory and especially to the subject of graph colorings: Chromatic Graph Theory. This map coloring problem would eventually acquire a name that would become known throughout the mathematical world. The Four Color Problem Can the countries of every map be colored with four or fewer colors so that every two countries with a common boundary are colored differently? 1
  • 17. 2 CHAPTER 0. THE ORIGIN OF GRAPH COLORINGS .................................................... ...................................... .............................................................. ...................................... ............................................................ ...................................... .............................................................. ...................................... ......................................................... ...................................... ............................................................................................................................................................................................................................................................................................................................ ............................................................... 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Venezuela Uruguay Falkland Islands Colombia Bolivia Chile Atlantic Ocean Brazil French Guiana Guyana Suriname Ecuador Peru Pacific Ocean Argentina Paraguay Figure 1: Map of South America Many of the concepts, theorems, and problems of Graph Theory lie in the shad- ows of the Four Color Problem. Indeed . . . Graph Theory is an area of mathematics whose past is always present. Since the maps we consider can be real or imagined, we can think of maps being divided into more general regions, rather than countries, states, provinces, or some other geographic entities. So just how did the Four Color Problem begin? It turns out that this question has a rather well-documented answer. On 23 October 1852, a student, namely Fred- erick Guthrie (1833–1886), at University College London visited his mathematics professor, the famous Augustus De Morgan (1806–1871), to describe an apparent mathematical discovery of his older brother Francis. While coloring the counties of a map of England, Francis Guthrie (1831–1899) observed that he could color them
  • 18. 3 with four colors, which led him to conjecture that no more than four colors would be needed to color the regions of any map. The Four Color Conjecture The regions of every map can be colored with four or fewer colors in such a way that every two regions sharing a common boundary are colored differently. Two years earlier, in 1850, Francis had earned a Bachelor of Arts degree from University College London and then a Bachelor of Laws degree in 1852. He would later become a mathematics professor himself at the University of Cape Town in South Africa. Francis developed a lifelong interest in botany and his extensive collection of flora from the Cape Peninsula would later be placed in the Guthrie Herbarium in the University of Cape Town Botany Department. Several rare species of flora are named for him. Francis Guthrie attempted to prove the Four Color Conjecture and although he thought he may have been successful, he was not completely satisfied with his proof. Francis discussed his discovery with Frederick. With Francis’ approval, Frederick mentioned the statement of this apparent theorem to Professor De Morgan, who expressed pleasure with it and believed it to be a new result. Evidently Frederick asked Professor De Morgan if he was aware of an argument that would establish the truth of the theorem. This led De Morgan to write a letter to his friend, the famous Irish mathemati- cian Sir William Rowan Hamilton (1805–1865) in Dublin. These two mathematical giants had corresponded for years, although apparently had met only once. De Mor- gan wrote (in part): My dear Hamilton: A student of mine asked me to day to give him a reason for a fact which I did not know was a fact – and do not yet. He says that if a figure be any how divided and the compartments differently coloured so that figures with any portion of common boundary lines are differently coloured – four colours may be wanted but not more – the following is his case in which four are wanted. ............................................................................................. ......................................................................................................................................................................... BA C DareA B C D names of colours Query cannot a necessity for five or more be invented . . . My pupil says he guessed it colouring a map of England . . .. The more I think of it the more evident it seems. If you retort with some very simple case which makes me out a stupid animal, I think I must do as the Sphynx did . . .
  • 19. 4 CHAPTER 0. THE ORIGIN OF GRAPH COLORINGS In De Morgan’s letter to Hamilton, he refers to the “Sphynx” (or Sphinx). While the Sphinx is a male statue of a lion with the head of a human in ancient Egypt which guards the entrance to a temple, the Greek Sphinx is a female creature of bad luck who sat atop a rock posing the following riddle to all those who pass by: What animal is that which in the morning goes on four feet, at noon on two, and in the evening upon three? Those who did not solve the riddle were killed. Only Oedipus (the title character in Oedipus Rex by Sophocles, a play about how people do not control their own destiny) answered the riddle correctly as “Man”, who in childhood (the morning of life) creeps on hands and knees, in manhood (the noon of life) walks upright, and in old age (the evening of life) walks with the aid of a cane. Upon learning that her riddle had been solved, the Sphinx cast herself from the rock and perished, a fate De Morgan had envisioned for himself if his riddle (the Four Color Problem) had an easy and immediate solution. In De Morgan’s letter to Hamilton, De Morgan attempted to explain why the problem appeared to be difficult. He followed this explanation by writing: But it is tricky work and I am not sure of all convolutions – What do you say? And has it, if true been noticed? Among Hamilton’s numerous mathematical accomplishments was his remark- able work with quaternions. Hamilton’s quaternions are a 4-dimensional system of numbers of the form a + bi + cj + dk, where a, b, c, d ∈ R and i2 = j2 = k2 = −1. When c = d = 0, these numbers are the 2-dimensional system of complex numbers; while when b = c = d = 0, these numbers are simply real numbers. Although it is commonplace for binary operations in algebraic structures to be commutative, such is not the case for products of quaternions. For example, i · j = k but j · i = −k. Since De Morgan had shown an interest in Hamilton’s research on quaternions as well as other subjects Hamilton had studied, it is likely that De Morgan expected an enthusiastic reply to his letter to Hamilton. Such was not the case, however. Indeed, on 26 October 1852, Hamilton gave a quick but unexpected response: I am not likely to attempt your “quaternion” of colours very soon. Hamilton’s response did nothing however to diminish De Morgan’s interest in the Four Color Problem. Since De Morgan’s letter to Hamilton did not mention Frederick Guthrie by name, there may be reason to question whether Frederick was in fact the student to whom De Morgan was referring and that it was Frederick’s older brother Francis who was the originator of the Four Color Problem. In 1852 Frederick Guthrie was a teenager. He would go on to become a distin- guished physics professor and founder of the Physical Society in London. An area that he studied is the science of thermionic emission – first reported by Frederick Guthrie in 1873. He discovered that a red-hot iron sphere with a positive charge
  • 20. 5 would lose its charge. This effect was rediscovered by the famous American inventor Thomas Edison early in 1880. It was during 1880 (only six years before Frederick died) that Frederick wrote: Some thirty years ago, when I was attending Professor De Morgan’s class, my brother, Francis Guthrie, who had recently ceased to attend then (and who is now professor of mathematics at the South African University, Cape Town), showed me the fact that the greatest necessary number of colours to be used in colouring a map so as to avoid identity colour in lineally contiguous districts is four. I should not be justified, after this lapse of time, in trying to give his proof, but the critical dia- gram was as in the margin. ........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... .............................................. ................................................. 1 4 32 With my brother’s permission I submitted the theorem to Professor De Morgan, who expressed himself very pleased with it; accepted it as new; and, as I am informed by those who subsequently attended his classes, was in the habit of acknowledging where he had got his infor- mation. If I remember rightly, the proof which my brother gave did not seem altogether satisfactory to himself; but I must refer to him those interested in the subject. . . .. The first statement in print of the Four Color Problem evidently occurred in an anonymous review written in the 14 April 1860 issue of the literary journal Athenaeum. Although the author of the review was not identified, De Morgan was quite clearly the writer. This review led to the Four Color Problem becoming known in the United States. The Four Color Problem came to the attention of the American mathematician Charles Sanders Peirce (1839–1914), who found an example of a map drawn on a torus (a donut-shaped surface) that required six colors. (As we will see in Chapter 5, there is an example of a map drawn on a torus that requires seven colors.) Peirce expressed great interest in the Four Color Problem. In fact, he visited De Morgan in 1870, who by that time was experiencing poor health. Indeed, De Morgan died the following year. Not only had De Morgan made little progress towards a solution of the Four Color Problem at the time of his death, overall interest in this problem had faded. While Peirce continued to attempt to solve the problem, De Morgan’s British acquaintances appeared to pay little attention to the problem – with at least one notable exception.
  • 21. 6 CHAPTER 0. THE ORIGIN OF GRAPH COLORINGS Arthur Cayley (1821–1895) graduated from Trinity College, Cambridge in 1842 and then received a fellowship from Cambridge, where he taught for four years. Afterwards, because of the limitations on his fellowship, he was required to choose a profession. He chose law, but only as a means to make money while he could continue to do mathematics. During 1849–1863, Cayley was a successful lawyer but published some 250 research papers during this period, including many for which he is well known. One of these was his pioneering paper on matrix algebra. Cayley was famous for his work on algebra, much of which was done with the British mathematician James Joseph Sylvester (1814–1897), a former student of De Morgan. In 1863 Cayley was appointed a professor of mathematics at Cambridge. Two years later, the London Mathematical Society was founded at University College London and would serve as a model for the American Mathematical Society, founded in 1888. De Morgan became the first president of the London Mathematical Society, followed by Sylvester and then Cayley. During a meeting of the Society on 13 June 1878, Cayley raised a question about the Four Color Problem that brought renewed attention to the problem: Has a solution been given of the statement that in colouring a map of a country, divided into counties, only four distinct colours are required, so that no two adjacent counties should be painted in the same colour? This question appeared in the Proceedings of the Society’s meeting. In the April 1879 issue of the Proceedings of the Royal Geographical Society, Cayley reported: I have not succeeded in obtaining a general proof; and it is worth while to explain wherein the difficulty consists. Cayley observed that if a map with a certain number of regions has been colored with four colors and a new map is obtained by adding a new region, then there is no guarantee that the new map can be colored with four colors – without first recoloring the original map. This showed that any attempted proof of the Four Color Conjecture using a proof by mathematical induction would not be straightforward. Another possible proof technique to try would be proof by contradiction. Applying this technique, we would assume that the Four Color Conjecture is false. This would mean that there are some maps that cannot be colored with four colors. Among the maps that require five or more colors are those with a smallest number of regions. Any one of these maps constitutes a minimum counterexample. If it could be shown that no minimum counterexample could exist, then this would establish the truth of the Four Color Conjecture. For example, no minimum counterexample M could possibly contain a region R surrounded by three regions R1, R2, and R3 as shown in Figure 2(a). In this case, we could shrink the region R to a point, producing a new map M′ with one less region. The map M′ can then be colored with four colors, only three of which are used to color R1, R2, and R3 as in Figure 2(b). Returning to the original map M, we see that there is now an available color for R as shown in Figure 2(c),
  • 22. 7 implying that M could be colored with four colors after all, thereby producing a contradiction. Certainly, if the map M contains a region surrounded by fewer than three regions, a contradiction can be obtained in the same manner. ...................................................................................................................................................................................................................................... ....................................................................................................................................................................................................................................... .................................................................................. ..................................................................................... .................................................................................. .................................................................................. ..................................................................................... ...................................... ...................................... ..................................................................................... R (a) (c) blue green red greenR3 (b) in M′ in Min M yellow blueR2 R1 yellow Figure 2: A region surrounded by three regions in a map Suppose, however, that the map M contained no region surrounded by three or fewer regions but did contain a region R surrounded by four regions, say R1, R2, R3, R4, as shown in Figure 3(a). If, once again, we shrink the region R to a point, producing a map M′ with one less region, then we know that M′ can be colored with four colors. If two or three colors are used to color R1, R2, R3, R4, then we can return to M and there is a color available for R. However, this technique does not work if the regions R1, R2, R3, R4 are colored with four distinct colors, as shown in Figure 3(b). ...................................................................................................................................................................................................................................... ...................................... .................................................................................................................................................................................... R R3 R1 green red R4 blue yellow R2 in M(a) (b) in M′ Figure 3: A region surrounded by four regions in a map What we can do in this case, however, is to determine whether the map M′ has a chain of regions, beginning at R1 and ending at R3, all of which are colored red or green. If no such chain exists, then the two colors of every red-green chain of regions beginning at R1 can be interchanged. We can then return to the map M, where the color red is now available for R. That is, the map M can be colored with four colors, producing a contradiction. But what if a red-green chain of regions beginning at R1 and ending at R3 exists? (See Figure 4, where r, b, g, y denote the colors red, blue, green, yellow.) Then interchanging the colors red and green offers no benefit to us. However, in this case, there can be no blue-yellow chain of regions, beginning at R2 and ending at R4. Then the colors of every blue-yellow chain of
  • 23. 8 CHAPTER 0. THE ORIGIN OF GRAPH COLORINGS regions beginning at R2 can be interchanged. Returning to M, we see that the color blue is now available for R, which once again says that M can be colored with four colors and produces a contradiction. .................... .......................... g rg r g g r R3 R1 R4 R2 b yg R r r ..................................................................................................................................................................................................................................................................................................................... ............................................................................................................................................................................................................................................................................................................... ......................... .............................................................................. ................................................................................................................................................................................................... .............................................................................................................................................................................................................................................................. ............. ..................................................................................................................................................................................................... ............................................................................................................................................................................................................................................................ ....................................................................................................................................................................................................................................................................................................................................................................... .......... .................................................................................................................................................................................................................................................. ................................... .................................................................................................................................................................................................................................................................................................................................................. ............................................................................................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................................................................... ................................................................................................................................................................................................... ......................................................................................................... ....................................................................... ........................................................................................................................................................................ Figure 4: A a red-green chain of regions from R1 to R3 It is possible to show (as we will see in Chapter 5) that a map may contain no region that is surrounded by four or fewer neighboring regions. Should this occur however, such a map must contain a region surrounded by exactly five neighboring regions. We mentioned that James Joseph Sylvester worked with Arthur Cayley and served as the second president of the London Mathematical Society. Sylvester, a superb mathematician himself, was invited to join the mathematics faculty of the newly founded Johns Hopkins University in Baltimore, Maryland in 1875. Included among his attempts to inspire more research at the university was his founding in 1878 of the American Journal of Mathematics, of which he held the position of editor-in-chief. While the goal of the journal was to serve American mathematicians, foreign submissions were encouraged as well, including articles from Sylvester’s friend Cayley. Among those who studied under Arthur Cayley was Alfred Bray Kempe (1849– 1922). Despite his great enthusiasm for mathematics, Kempe took up a career in the legal profession. Kempe was present at the meeting of the London Mathematical Society in which Cayley had inquired about the status of the Four Color Problem. Kempe worked on the problem and obtained a solution in 1879. Indeed, on 17 July 1879 a statement of Kempe’s accomplishment appeared in the British journal Nature, with the complete proof published in Volume 2 of Sylvester’s American Journal of Mathematics. Kempe’s approach for solving the Four Color Problem essentially followed the technique described earlier. His technique involved locating a region R in a map M such that R is surrounded by five or fewer neighboring regions and showing that for every coloring of M (minus the region R) with four colors, there is a coloring of the entire map M with four colors. Such an argument would show that M could not be a minimum counterexample. We saw how such a proof would proceed if R were
  • 24. 9 surrounded by four or fewer neighboring regions. This included looking for chains of regions whose colors alternate between two colors and then interchanging these colors, if appropriate, to arrive at a coloring of the regions of M (minus R) with four colors so that the neighboring regions of R used at most three of these colors and thereby leaving a color available for R. In fact, these chains of regions became known as Kempe chains, for it was Kempe who originated this idea. There was one case, however, that still needed to be resolved, namely the case where no region in the map was surrounded by four or fewer neighboring regions. As we noted, the map must then contain some region R surrounded by exactly five neighboring regions. At least three of the four colors must be used to color the five neighboring regions of R. If only three colors are used to color these five regions, then a color is available for R. Hence we are left with the single situation in which all four colors are used to color the five neighboring regions surrounding R (see Figure 5), where once again r, b, g, y indicate the colors red, blue, green, yellow. ........................................................... ...................................... ........................................................................................................... ............................................................ ...................................... .................................................................... ...................................... ............................................................... ...................................... .............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ...................................................... ............................................................. ...................................................................... ....................................................................... ........................................................................................ ............................................................................................................................................................................................................................................................................................................................................................................................................................................................... ...................................................................................................................................................................................................................................................................................... .......................................................................................................................................................................................................................................... ..................................................................................................................................................................................................................................... R g y b r b R4 R2 R5 R3 R1 Figure 5: The final case in Kempe’s solution of the Four Color Problem Let’s see how Kempe handled this final case. Among the regions adjacent to R, only the region R1 is colored yellow. Consider all the regions of the map M that are colored either yellow or red and that, beginning at R1, can be reached by an alternating sequence of neighboring yellow and red regions, that is, by a yellow-red Kempe chain. If the region R3 (which is the neighboring region of R colored red) cannot be reached by a yellow-red Kempe chain, then the colors yellow and red can be interchanged for all regions in M that can be reached by a yellow-red Kempe chain beginning at R1. This results in a coloring of all regions in M (except R) in which neighboring regions are colored differently and such that each neighboring region of R is colored red, blue, or green. We can then color R yellow to arrive at a 4-coloring of the entire map M. From this, we may assume that the region R3 can be reached by a yellow-red Kempe chain beginning at R1. (See Figure 6.) Let’s now look at the region R5, which is colored green. We consider all regions of M colored green or red and that, beginning at R5, can be reached by a green-red Kempe chain. If the region R3 cannot be reached by a green-red Kempe chain that begins at R5, then the colors green and red can be interchanged for all regions in M
  • 25. 10 CHAPTER 0. THE ORIGIN OF GRAPH COLORINGS .............................................................. ...................................... ............................................................. ...................................... .................................................................. ...................................... .............................................................. ...................................... ..................................................................................................... .............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ...................................................... ............................................................. ...................................................................... ....................................................................... ........................................................................................ ............................................................................................................................................................................................................................................................................................................................................................................................................................................................... ...................................................................................................................................................................................................................................................................................... ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ..................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................................................................................................................................................................................................................. ....................................................................................................................................................................................................................................................................................................................................... ............................................................................................................................................................................................................................................................................................... ............................................................................................ ............................................................................................................................................................................................................................... R r r y y y r y r b g b R3 R4 R5 R1 R2 Figure 6: A yellow-red Kempe chain in the map M that can be reached by a green-red Kempe chain beginning at R5. Upon doing this, a 4-coloring of all regions in M (except R) is obtained, in which each neighboring region of R is colored red, blue, or yellow. We can then color R green to produce a 4-coloring of the entire map M. We may therefore assume that R3 can be reached by a green-red Kempe chain that begins at R5. (See Figure 7.) .............................................................................................. ................................................... ...................................... .......................................................................... ...................................... ............................................................. ...................................... ............................................................................................................... ...................................... 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............................................................................................................................................................. y y g r r r r r y r g r g g b b gy RR2 r R4 R3 R5 R1 Figure 7: Yellow-red and green-red Kempe chains in the map M Because there is a ring of regions consisting of R and a green-red Kempe chain, there cannot be a blue-yellow Kempe chain in M beginning at R4 and ending at R1. In addition, because there is a ring of regions consisting of R and a yellow-red Kempe chain, there is no blue-green Kempe chain in M beginning at R2 and ending at R5. Hence we interchange the colors blue and yellow for all regions in M that can be reached by a blue-yellow Kempe chain beginning at R4 and interchange the colors
  • 26. 11 blue and green for all regions in M that can be reached by a blue-green Kempe chain beginning at R2. Once these two color interchanges have been performed, each of the five neighboring regions of R is colored red, yellow, or green. Then R can be colored blue and a 4-coloring of the map M has been obtained, completing the proof. As it turned out, the proof given by Kempe contained a fatal flaw, but one that would go unnoticed for a decade. Despite the fact that Kempe’s attempted proof of the Four Color Problem was erroneous, he made a number of interesting observations in his article. He noticed that if a piece of tracing paper was placed over a map and a point was marked on the tracing paper over each region of the map and two points were joined by a line segment whenever the corresponding regions had a common boundary, then a diagram of a “linkage” was produced. Furthermore, the problem of determining whether the regions of the map can be colored with four colors so that neighboring regions are colored differently is the same problem as determining whether the points in the linkage can be colored with four colors so that every two points joined by a line segment are colored differently. (See Figure 8.) ................................................................................................................................................................................................................................................................................................................................. .......................................................................................................................................................................................................................... ............................................................................................................................................................................................. ............................................................................................................................................................................................. 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ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp s pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppps s s s s s s s s Figure 8: A map and corresponding planar graph In 1878 Sylvester referred to a linkage as a graph and it is this terminology that became accepted. Later it became commonplace to refer to the points and lines of a linkage as the vertices and edges of the graph (with “vertex” being the singular of “vertices”). Since the graphs constructed from maps in this manner (referred to as the dual graph of the map) can themselves be drawn in the plane without two edges (line segments) intersecting, these graphs were called planar graphs. A planar graph that is actually drawn in the plane without any of its edges intersecting is called a plane graph. In terms of graphs, the Four Color Conjecture could then be restated. The Four Color Conjecture The vertices of every planar graph can be colored with four or fewer colors in such a way that every two vertices joined by an edge are colored differently. Indeed, the vast majority of this book will be devoted to coloring graphs (not coloring maps) and, in fact, to coloring graphs in general, not only planar graphs.
  • 27. 12 CHAPTER 0. THE ORIGIN OF GRAPH COLORINGS The colouring of abstract graphs is a generalization of the colouring of maps, and the study of the colouring of abstract graphs . . . opens a new chapter in the combinatorial part of mathematics. Gabriel Andrew Dirac (1951) For the present, however, we continue our discussion in terms of coloring the regions of maps. Kempe’s proof of the theorem, which had become known as the Four Color Theorem, was accepted both within the United States and England. Arthur Cayley had accepted Kempe’s argument as a valid proof. This led to Kempe being elected as a Fellow of the Royal Society in 1881. The Four Color Theorem The regions of every map can be colored with four or fewer colors so that every two adjacent regions are colored differently. Among the many individuals who had become interested in the Four Color Problem was Charles Lutwidge Dodgson (1832–1898), an Englishman with a keen interest in mathematics and puzzles. Dodgson was better known, however, under his pen-name Lewis Carroll and for his well-known books Alice’s Adventures in Wonderland and Through the Looking-Glass and What Alice Found There. Another well-known individual with mathematical interests, but whose primary occupation was not that of a mathematician, was Frederick Temple (1821–1902), Bishop of London and who would later become the Archbishop of Canterbury. Like Dodgson and others, Temple had a fondness for puzzles. Temple showed that it was impossible to have five mutually neighboring regions in any map and from this concluded that no map required five colors. Although Temple was correct about the non-existence of five mutually neighboring regions in a map, his conclusion that this provided a proof of the Four Color Conjecture was incorrect. There was historical precedence about the non-existence of five mutually ad- jacent regions in any map. In 1840 the famous German mathematician August M¨obius (1790–1868) reportedly stated the following problem, which was proposed to him by the philologist Benjamin Weiske (1748–1809). Problem of Five Princes There was once a king with five sons. In his will, he stated that after his death his kingdom should be divided into five regions in such a way that each region should have a common boundary with the other four. Can the terms of the will be satisfied? As we noted, the conditions of the king’s will cannot be met. This problem illustrates M¨obius’ interest in topology, a subject of which M¨obius was one of the early pioneers. In a memoir written by M¨obius and only discovered after his death, he discussed properties of one-sided surfaces, which became known as M¨obius strips (even though it was determined that Johann Listing (1808–1882) had discovered these earlier).
  • 28. 13 In 1885 the German geometer Richard Baltzer (1818–1887) also lectured on the non-existence of five mutually adjacent regions. In the published version of his lecture, it was incorrectly stated that the Four Color Theorem followed from this. This error was repeated by other writers until the famous geometer Harold Scott MacDonald Coxeter (1907–2003) corrected the matter in 1959. Mistakes concerning the Four Color Problem were not limited to mathematical errors however. Prior to establishing Francis Guthrie as the true and sole originator of the Four Color Problem, it was often stated in print that cartographers were aware that the regions of every map could be colored with four or less colors so that adjacent regions are colored differently. The well-known mathematical historian Kenneth O. May (1915–1977) investigated this claim and found no justification to it. He conducted a study of atlases in the Library of Congress and found no evidence of attempts to minimize the number of colors used in maps. Most maps used more than four colors and even when four colors were used, often less colors could have been used. There was never a mention of a “four color theorem”. Another mathematician of note around 1880 was Peter Guthrie Tait (1831– 1901). In addition to being a scholar, he was a golf enthusiast. His son Frederick Guthrie Tait was a champion golfer and considered a national hero in Scotland. The first golf biography ever written was about Frederick Tait. Indeed, the Freddie Tait Golf Week is held every year in Kimberley, South Africa to commemorate his life as a golfer and soldier. He was killed during the Anglo-Boer War of 1899–1902. Peter Guthrie Tait had heard of the Four Color Conjecture through Arthur Cayley and was aware of Kempe’s solution. He felt that Kempe’s solution of the Four Color Problem was overly long and gave several shorter solutions of the problem, all of which turned out to be incorrect. Despite this, one of his attempted proofs contained an interesting and useful idea. A type of map that is often encountered is a cubic map, in which there are exactly three boundary lines at each meeting point. In fact, every map M that has no region completely surrounded by another region can be converted into a cubic map M′ by drawing a circle about each meeting point in M′ and creating new meeting points and one new region (see Figure 9). If the map M′ can be colored with four colors, then so can M. .............................................................................................................................................................................................................................................................................................................................................................................. ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ ...................................... .............................................................................................................................................................................................. ...................................................................................................................................................................................................... .................................................................................................. ...................................... ............................................. ...................................... .......................................... ........................................ .................................................................................................... in M in M′ Figure 9: Converting a map into a cubic map Tait’s idea was to consider coloring the boundary lines of cubic maps. In fact, he stated as a lemma that:
  • 29. 14 CHAPTER 0. THE ORIGIN OF GRAPH COLORINGS The boundary lines of every cubic map can always be colored with three colors so that the three lines at each meeting point are colored differently. Tait also mentioned that this lemma could be easily proved and showed how the lemma could be used to prove the Four Color Theorem. Although Tait was correct that this lemma could be used to to prove the Four Color Theorem, he was incorrect when he said that the lemma could be easily proved. Indeed, as it turned out, this lemma is equivalent to the Four Color Theorem and, of course, is equally difficult to prove. (We will discuss Tait’s coloring of the boundary lines of cubic maps in Chapter 10.) The next important figure in the history of the Four Color Problem was Percy John Heawood (1861–1955), who spent the period 1887–1939 as a lecturer, professor, and vice-chancellor at Durham College in England. When Heawood was a student at Oxford University in 1880, one of his teachers was Professor Henry Smith who spoke often of the Four Color Problem. Heawood read Kempe’s paper and it was he who discovered the serious error in the proof. In 1889 Heawood wrote a paper of his own, published in 1890, in which he presented the map shown in Figure 10. r g b y y b ........................................................................................................ ........................................................................................................................................................................................................................................................................................................................................................................................................................................................... ............................................................................ ................................................................................................. ........................................................................................................ 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Figure 10: Heawood’s counterexample to Kempe’s proof In the Heawood map, two of the five neighboring regions surrounding the uncol- ored region R are colored red; while for each of the colors blue, yellow, and green, there is exactly one neighboring region of R with that color. According to Kempe’s argument, since blue is the color of the region that shares a boundary with R as well as with the two neighboring regions of R colored red, we are concerned with whether this map contains a blue-yellow Kempe chain between two neighboring re- gions of R as well as a blue-green Kempe chain between two neighboring regions of